Obtaining Impulse Response & Frequency Response of LTI System

[coolrp]

Homework Statement

An LTI discrete-time system has frequency response given by

(a) Use one of the above forms of the frequency response to obtain an equation for the
impulse response h[n] of the system.

(b) From the frequency response, determine the difference equation that is satisfied by
the input x[n] and the output y[n] of the system.

(c) If the input to this system is
x[n]=4+2cos(won) for oo<n<oo,
for what value of wo will the output be of the form
y[n] =A constant
for -00 < n < oo? What is the constant A?


2. The attempt at a solution
(a) i found to be
h[n] = (0.8)^n u[n] + (0.8)^n-2 u[n-2]

Can anyone help me with the rest of the two parts?

 

[KataKoniK]

I can assist you with the remaining parts of this problem. Let's start with part (b).

To determine the difference equation satisfied by the input x[n] and output y[n], we can use the formula for the frequency response in the form of H(e^jw) = Y(e^jw)/X(e^jw). This means that the output y[n] can be expressed as the input x[n] multiplied by the frequency response H(e^jw).

Therefore, the difference equation can be written as y[n] = h[n]*x[n]. Substituting the frequency response given in the problem, we get y[n] = (0.8)^n u[n] + (0.8)^n-2 u[n-2] * x[n].

Moving on to part (c), we need to determine the value of wo for which the output y[n] will be of the form y[n] = A constant. From the given input x[n], we can see that it is a sinusoidal signal with a frequency of wo. Therefore, we need to find the value of wo for which the frequency response of the system will be zero.

To do this, we can set H(e^jw) = 0 and solve for w. From the given frequency response, we get w = -pi/2. This means that the value of wo that will result in a constant output is wo = pi/2.

Substituting this value of wo in the given input x[n], we get x[n] = 4 + 2cos(pi/2*n) = 4 + 2*(-1)^n.

Therefore, the output y[n] will be a constant for -00 < n < oo, with the value of A = 4.

I hope this helps you with the remaining parts of the problem. Let me know if you have any further questions.